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Roots and Powers in Regular Languages: Recognizing Nonregular Properties by Finite Automata




TekijätFrei Fabian, Hromkovic Juraj, Karhumäki Juhani

KustantajaIOS PRESS

Julkaisuvuosi2020

JournalFundamenta Informaticae

Tietokannassa oleva lehden nimiFUNDAMENTA INFORMATICAE

Lehden akronyymiFUND INFORM

Vuosikerta175

Numero1-4

Aloitussivu173

Lopetussivu185

Sivujen määrä13

ISSN0169-2968

eISSN1875-8681

DOIhttps://doi.org/10.3233/FI-2020-1952


Tiivistelmä
It is well known that the set of powers of any given order, for example squares, in a regular language need not be regular. Nevertheless, finite automata can identify them via their roots. More precisely, we recall that, given a regular language L, the set of square roots of L is regular. The same holds true for the nth roots for any n and for the set of all nontrivial roots; we give a concrete construction for all of them.Using the above result, we obtain decision algorithms for many natural problems on powers. For example, it is decidable, given two regular languages, whether they contain the same number of squares at each length. Finally, we give an exponential lower bound on the size of automata identifying powers in regular languages. Moreover, we highlight interesting behavior differences between taking fractional powers of regular languages and taking prefixes of a fractional length. Indeed, fractional roots in a regular language can typically not be identified by finite automata.



Last updated on 2024-26-11 at 18:25